8,168 research outputs found
A normal form algorithm for the Brieskorn lattice
This article describes a normal form algorithm for the Brieskorn lattice of
an isolated hypersurface singularity. It is the basis of efficient algorithms
to compute the Bernstein-Sato polynomial, the complex monodromy, and
Hodge-theoretic invariants of the singularity such as the spectral pairs and
good bases of the Brieskorn lattice. The algorithm is a variant of Buchberger's
normal form algorithm for power series rings using the idea of partial standard
bases and adic convergence replacing termination.Comment: 23 pages, 1 figure, 4 table
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
Straightening rule for an -truncated polynomial ring
We consider a certain quotient of a polynomial ring categorified by both the
isomorphic Green rings of the symmetric groups and Schur algebras generated by
the signed Young permutation modules and mixed powers respectively. They have
bases parametrised by pairs of partitions whose second partitions are multiples
of the odd prime the characteristic of the underlying field. We provide an
explicit formula rewriting a signed Young permutation module (respectively,
mixed power) in terms of signed Young permutation modules (respectively, mixed
powers) labelled by those pairs of partitions. As a result, for each partition
, we discovered the number of compositions such that
can be rearranged to and whose partial sums of are not
divisible by
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